Mathematics: Difference between revisions - Wikipedia

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Before the [[Renaissance]], mathematics was divided into two main areas: [[arithmetic]], regarding the manipulation of numbers, and [[geometry]], regarding the study of shapes.<ref>{{cite book |last=Bell |first=E. T. |author-link=Eric Temple Bell |year=1945 |orig-date=1940 |chapter=General Prospectus |title=The Development of Mathematics |edition=2nd |isbn=978-0-486-27239-9 |lccn=45010599 |oclc=523284 |page=3 |publisher=Dover Publications |quote=... mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry.}}</ref> Some types of [[pseudoscience]], such as [[numerology]] and [[astrology]], were not then clearly distinguished from mathematics.<ref>{{cite book |last=Tiwari |first=Sarju |year=1992 |chapter=A Mirror of Civilization |title=Mathematics in History, Culture, Philosophy, and Science |edition=1st |page=27 |publisher=Mittal Publications |publication-place=New Delhi, India |isbn=978-81-7099-404-6 |lccn=92909575 |oclc=28115124 |quote=It is unfortunate that two curses of mathematics--Numerology and Astrology were also born with it and have been more acceptable to the masses than mathematics itself.}}</ref>

During the Renaissance, two more areas appeared. [[Mathematical notation]] led to [[algebra]] which, roughly speaking, consists of the study and the manipulation of [[formula]]s. [[Calculus]], consisting of the two subfields ''[[differential calculus]]'' and ''[[integral calculus]]'', is the study of [[continuous functions]], which model the typically [[Nonlinear system|nonlinear relationships]] between varying quantities, as represented by [[variable (mathematics)|variables]]. This division into four main areas{{endashemdash}}arithmetic, geometry, algebra, calculus<ref>{{cite book |last=Restivo |first=Sal |author-link=Sal Restivo |editor-last=Bunge |editor-first=Mario |editor-link=Mario Bunge |year=1992 |chapter=Mathematics from the Ground Up |title=Mathematics in Society and History |page=14 |series=Episteme |volume=20 |publisher=[[Kluwer Academic Publishers]] |isbn=0-7923-1765-3 |lccn=25709270 |oclc=92013695}}</ref>{{endashemdash}}endured until the end of the 19th century. Areas such as [[celestial mechanics]] and [[solid mechanics]] were then studied by mathematicians, but now are considered as belonging to physics.<ref>{{cite book |last=Musielak |first=Dora |author-link=Dora Musielak |year=2022 |title=Leonhard Euler and the Foundations of Celestial Mechanics |series=History of Physics |publisher=[[Springer International Publishing]] |doi=10.1007/978-3-031-12322-1 |isbn=978-3-031-12321-4 |s2cid=253240718 |issn=2730-7549 |eissn=2730-7557 |oclc=1332780664}}</ref> The subject of [[combinatorics]] has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.<ref>{{cite journal |date=May 1979 |last=Biggs |first=N. L. |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0 |doi-access=free |issn=0315-0860 |eissn=1090-249X |lccn=75642280 |oclc=2240703}}</ref>

At the end of the 19th century, the [[foundational crisis in mathematics]] and the resulting systematization of the [[axiomatic method]] led to an explosion of new areas of mathematics.<ref name=Warner_2013>{{cite web |last=Warner |first=Evan |title=Splash Talk: The Foundational Crisis of Mathematics |publisher=[[Columbia University]] |url=https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf |url-status=dead |archive-url=https://web.archive.org/web/20230322165544/https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf |archive-date=March 22, 2023 |access-date=February 3, 2024}}</ref><ref name="Kleiner_1991" /> The 2020 [[Mathematics Subject Classification]] contains no less than {{em|sixty-three}} first-level areas.<ref>{{cite journal |last1=Dunne |first1=Edward |last2=Hulek |first2=Klaus |author2-link=Klaus Hulek |date=March 2020 |title=Mathematics Subject Classification 2020 |journal=Notices of the American Mathematical Society |volume=67 |issue=3 |pages=410–411 |doi=10.1090/noti2052 |doi-access=free |issn=0002-9920 |eissn=1088-9477 |lccn=sf77000404 |oclc=1480366 |url=https://www.ams.org/journals/notices/202003/rnoti-p410.pdf |url-status=live |archive-url=https://web.archive.org/web/20210803203928/https://www.ams.org/journals/notices/202003/rnoti-p410.pdf |archive-date=August 3, 2021 |access-date=February 3, 2024 |quote=The new MSC contains 63 two-digit classifications, 529 three-digit classifications, and 6,006 five-digit classifications.}}</ref> Some of these areas correspond to the older division, as is true regarding [[number theory]] (the modern name for [[higher arithmetic]]) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as [[mathematical logic]] and [[foundations of mathematics|foundations]].<ref name=MSC>{{cite web |url=https://zbmath.org/static/msc2020.pdf |title=MSC2020-Mathematics Subject Classification System |website=zbMath |publisher=Associate Editors of Mathematical Reviews and zbMATH |url-status=live |archive-url=https://web.archive.org/web/20240102023805/https://zbmath.org/static/msc2020.pdf |archive-date=January 2, 2024 |access-date=February 3, 2024}}</ref>

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| url-status=live

}}</ref><ref>{{cite conference | title=Grand Challenges, High Performance Computing, and Computational Science | last1=Johnson | first1=Gary M. | last2=Cavallini | first2=John S. | conference=Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage | date=September 1991 | page=28 |lccn=91018998 |publisher=World Scientific | editor1-first=Kang Hoh | editor1-last=Phua | editor2-first=Kia Fock | editor2-last=Loe | url={{GBurl|id=jYNIDwAAQBAJ|p=28}} | access-date=November 13, 2022 }}</ref> [[Numerical analysis]] studies methods for problems in [[analysis (mathematics)|analysis]] using [[functional analysis]] and [[approximation theory]]; numerical analysis broadly includes the study of [[approximation]] and [[discretization]] with special focus on [[rounding error]]s.<ref>{{cite book |last=Trefethen |first=Lloyd N. |author-link=Lloyd N. Trefethen |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor2-link=June Barrow-Green |editor3-last=Leader |editor3-first=Imre |editor3-link=Imre Leader |year=2008 |chapter=Numerical Analysis |pages=604–615 |title=The Princeton Companion to Mathematics |publisher=[[Princeton University Press]] |isbn=978-0-691-11880-2 |lccn=2008020450 |mr=2467561 |oclc=227205932 |url=http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |url-status=live |archive-url=https://web.archive.org/web/20230307054158/http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |archive-date=March 7, 2023 |access-date=February 15, 2024}}</ref> Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-[[numerical linear algebra|matrix]]-and-[[graph theory]]. Other areas of computational mathematics include [[computer algebra]] and [[symbolic computation]].

== History ==

{{Main|History of mathematics}}

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| volume=56 | issue=1 | date=January 1949 | pages=35–56

| doi=10.2307/2304570 | jstor=2304570

}}</ref> In the early 20th century, [[Kurt Gödel]] transformed mathematics by publishing [[Gödel's incompleteness theorems|his incompleteness theorems]], which show in part that any consistent axiomatic system{{emdash}}if powerful enough to describe arithmetic{{emdash}}will contain true propositions that cannot be proved.<ref name=Raatikainen_2005 />

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and [[science]], to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January&nbsp;2006 issue of the ''[[Bulletin of the American Mathematical Society]]'', "The number of papers and books included in the ''[[Mathematical Reviews]]'' database since 1940 (the first year of operation of MR) is now more than 1.9&nbsp;million, and more than 75&nbsp;thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."{{sfn|Sevryuk|2006|pp=101–109}}

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Mathematization of the social sciences is not without risk. In the controversial book ''[[Fashionable Nonsense]]'' (1997), [[Alan Sokal|Sokal]] and [[Jean Bricmont|Bricmont]] denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.<ref>{{cite book|last=Sokal|first=Alan|url=https://archive.org/details/fashionablenonse00soka|title=Fashionable Nonsense|author2=Jean Bricmont|publisher=Picador|year=1998|isbn=978-0-312-19545-8|location=New York|oclc=39605994|author-link=Alan Sokal|author2-link=Jean Bricmont}}</ref> The study of [[complex systems]] (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.<ref>{{Cite web|url=https://www.factcheck.org/2023/01/bidens-misleading-unemployment-statistic/|title=Biden's Misleading Unemployment Statistic – FactCheck.org}}</ref><ref>{{Cite web|url=https://www.minneapolisfed.org/article/2010/modern-macroeconomic-models-as-tools-for-economic-policy|title=Modern Macroeconomic Models as Tools for Economic Policy &#124; Federal Reserve Bank of Minneapolis|website=www.minneapolisfed.org}}</ref>

== Relationship with astrology and esotericism ==

Some renowned mathematicians have also been considered to be renowned [[astrologists]]; for example, [[Ptolemy]], Arab astronomers, [[Regiomantus]], [[Gerolamo Cardano|Cardano]], [[Kepler]], or [[John Dee]]. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, [[Theodor Zwinger]] wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences [of stars]" to foresee their "conjunctions and oppositions".<ref>{{Cite book |last=Beaujouan |first=Guy |url={{GBurl|id=92n7ZE8Iww8C|p=130}} |title=Comprendre et maîtriser la nature au Moyen Age: mélanges d'histoire des sciences offerts à Guy Beaujouan |date=1994 |publisher=Librairie Droz |isbn=978-2-600-00040-6 |page=130 |language=fr |access-date=January 3, 2023 }}</ref> As of 2023, astrology is no longer considered a science, but [[pseudoscience]].<ref>{{Cite web |title=L'astrologie à l'épreuve : ça ne marche pas, ça n'a jamais marché ! / Afis Science – Association française pour l'information scientifique |url=https://www.afis.org/L-astrologie-a-l-epreuve-ca-ne-marche-pas-ca-n-a-jamais-marche |access-date=December 28, 2022 |website=Afis Science – Association française pour l’information scientifique |language=fr |archive-date=January 29, 2023 |archive-url=https://web.archive.org/web/20230129204349/https://www.afis.org/L-astrologie-a-l-epreuve-ca-ne-marche-pas-ca-n-a-jamais-marche |url-status=live }}</ref>

== Philosophy ==

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{{Main|Popular mathematics}}Popular mathematics is the act of presenting mathematics without technical terms.<ref>{{Cite conference |last=Kissane |first=Barry |date=July 2009 |title=Popular mathematics |url=https://researchrepository.murdoch.edu.au/id/eprint/6242/ |conference=22nd Biennial Conference of The Australian Association of Mathematics Teachers |location=Fremantle, Western Australia |publisher=Australian Association of Mathematics Teachers |pages=125–126 |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054610/https://researchrepository.murdoch.edu.au/id/eprint/6242/ |url-status=live }}</ref> Presenting mathematics may be hard since the general public suffers from [[mathematical anxiety]] and mathematical objects are highly abstract.<ref>{{Cite book |last=Steen |first=L. A. |url={{GBurl|id=-d3TBwAAQBAJ|dq="popular mathematics" analogies|p=2}} |title=Mathematics Today Twelve Informal Essays |date=2012|publisher=Springer Science & Business Media |isbn=978-1-4613-9435-8 |page=2 |language=en |access-date=January 3, 2023 }}</ref> However, popular mathematics writing can overcome this by using applications or cultural links.<ref>{{Cite book |last=Pitici |first=Mircea |url={{GBurl|id=9nGQDQAAQBAJ|dq="popular mathematics" analogies|p=331}} |title=The Best Writing on Mathematics 2016 |date=2017|publisher=Princeton University Press |isbn=978-1-4008-8560-2 |language=en |access-date=January 3, 2023 }}</ref> Despite this, mathematics is rarely the topic of popularization in printed or televised media.

=== Awards and prize problems ===

{{Main category|Mathematics awards}}

[[File:FieldsMedalFront.jpg|thumb|The front side of the [[Fields Medal]] with an illustration of the Greek [[polymath]] [[Archimedes]]]]